\subsubsection{Bit Depth}
\textit{Bit depth} is a term commonly used in digital audio. It's the spectrum of bits that is available to determine the resolution of processed data. The higher the bit depth is (e.g. 24 bit), the more flexible and accurate the processing will be. On the contrary, having low bit depth (e.g. 16 bit) indicates the output will be more inaccurate, in addition to some frequencies being consequently lost in the process \cite{bit.depth1}.

With a 16-bit sequence, there are 65,536 possible levels. With every additional bit, there is double the amount of levels. When there is a 24-bit process or piece of 24-bit hardware, there are 16,777,216 available levels. 24-bit is the recommended bit depth for capturing the full dynamic range \cite{bit.depth2}.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.60\textwidth]{images/TheoryDesign/BitGraph.png}
\caption{A visual illustration of the different bit depth \cite{bitdepth}.}
\label{fig:bitdepth}
\end{figure}

\subsubsection{Quantization}
When converting analog signals to its digital counterpart, \textit{quantization} is the process of converting the continuous signal into a discrete signal. These samples are based on the nearest rounded up number in the sample. Thus, the accuracy of the conversion is dependent on the bit depth. 

\subsubsection{Nyquist Theorem}
\textit{The Nyquist Theorem}, also known as the \textit{sampling theorem}, is a concept that is involved in the conversion of analog-to-digital signals and anti-aliasing. In order for analog-to-digital conversion (ADC) to result in a near perfect copy of the original signal, the sampling of the analog signal must be done with a minimum frequency (see figure \ref{fig:Sampling}. 

According to the Nyquist Theorem, the sampling rate must be at least twice the maximum
($2fmax$) analog input in order to extract all of the information from the bandwidth and accurately correspond to the input's acoustic energy. Consequently, there is a risk of some of the higher frequency components of the analog input signal not being correctly represented in the digitized output, if the sampling rate is less than $2fmax$. Thus, if one wishes to process signals from 20 Hz to 20,000 Hz (e.g. from analog input to a CD), the frequency must be sampled at a rate of 40,000 Hz to reproduce a 20,000 Hz signal. By default, the CD standard is to sample 44,100 times per second, or 44.1 kHz \cite{nyquist1}.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.50\textwidth]{images/sa4.jpg}
\caption{A comparison between an accurately sampled and under-sampled signal \cite{sampling}.}
\label{fig:Sampling}
\end{figure}

Furthermore, when a digital signal is re-converted to analog form by a digital-to-analog converter, frequency components that were not present in the original analog input can be generated. This undesirable condition is a form of distortion called aliasing \cite{nyquist2}.

\subsubsection{Anti Aliasing}
\textit{Anti-aliasing} is the process of smoothing erroneous frequencies or values after performing sampling. In the context of images, aliased images will appear jagged and develop sharp color transitions; while for sound, aliased sounds are manifested as inaccurate frequencies or buzzes (see figure \ref{fig:antialias}). Anti-aliasing itself is a process that is applied when the data is sampled, therefore, accurate frequency measurements of the original signal must be obtained before sampling takes place \cite{antialiasing1}.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.50\textwidth]{images/TheoryDesign/antialias}
\caption{Top: aliasing Bottom: anti-aliasing \cite{antialias}.}
\label{fig:antialias}
\end{figure}

